How to make a rigid body pass through only certain other rigid bodies?
Rigid bodies collisions do not allow the kind of control you describe so, instead, a different approach is required.

First model the system using non-overlapping rigid bodies - create the rigid bodies and enable collision and use Rigid Body Constraints to set up the relationships between them.



The constraints can be easily created using the 'Connect' button in the Physics tab of the 3d view Tool Shelf (T). However, for each connection you need to uncheck the Disable Collisions property in the Rigid Body Constraints panel so that the collisions remain active.




To add the overlapping elements, create new objects and size and position them as desired and parent each one to the relevant rigid body (select the new object, hold Shift and select the associated rigid body, then press Ctrl+P and select 'Object') - they will now follow the motion of the associated rigid body.

Since the overlapping elements are no longer rigid bodies they do not affect the collision (and so can overlap) while the (non-overlapping) rigid bodies can handle the physics. You can 'hide' the rigid body 'skeleton' by moving it to another layer if desired - or simply set it not to render (click the camera icon against each object in the Outliner).



To take this a step further you could parent bones to the rigid bodies and have those deform a mesh that would act as a 'skin' to the rigid body skeleton. This is more complicated but would produce much more convincing joint motion for 'organic' characters.